Copied to
clipboard

G = GL2(𝔽3)⋊D5order 480 = 25·3·5

1st semidirect product of GL2(𝔽3) and D5 acting via D5/C5=C2

Aliases: Dic5.2S4, GL2(𝔽3)⋊1D5, SL2(𝔽3)⋊1D10, C2.7(D5×S4), C10.4(C2×S4), Q8⋊D154C2, Q8.4(S3×D5), (C5×Q8).4D6, C51(C4.3S4), Q82D53S3, Dic5.A43C2, (C5×GL2(𝔽3))⋊1C2, (C5×SL2(𝔽3))⋊1C22, SmallGroup(480,970)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — GL2(𝔽3)⋊D5
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — Dic5.A4 — GL2(𝔽3)⋊D5
 Lower central C5×SL2(𝔽3) — GL2(𝔽3)⋊D5
 Upper central C1 — C2

Generators and relations for GL2(𝔽3)⋊D5
G = < a,b,c,d,e,f | a4=c3=d2=e5=f2=1, b2=a2, bab-1=faf=dbd=a-1, cac-1=ab, dad=fbf=a2b, ae=ea, cbc-1=a, be=eb, dcd=c-1, ce=ec, fcf=ac, de=ed, df=fd, fef=e-1 >

Subgroups: 818 in 84 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, C23, D5, C10, C10, C12, D6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, SL2(𝔽3), D12, C5×S3, D15, C30, C8⋊C22, C52C8, C40, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, GL2(𝔽3), GL2(𝔽3), C4.A4, C3×Dic5, S3×C10, D30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, C4.3S4, C5⋊D12, C5×SL2(𝔽3), D40⋊C2, C5×GL2(𝔽3), Q8⋊D15, Dic5.A4, GL2(𝔽3)⋊D5
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, C2×S4, S3×D5, C4.3S4, D5×S4, GL2(𝔽3)⋊D5

Character table of GL2(𝔽3)⋊D5

 class 1 2A 2B 2C 2D 3 4A 4B 5A 5B 6 8A 8B 10A 10B 10C 10D 12A 12B 15A 15B 20A 20B 30A 30B 40A 40B 40C 40D size 1 1 12 30 60 8 6 10 2 2 8 12 60 2 2 24 24 40 40 16 16 12 12 16 16 12 12 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 -1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 -1 -1 1 1 -1 1 1 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 -1 1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 2 2 0 -2 0 -1 2 -2 2 2 -1 0 0 2 2 0 0 1 1 -1 -1 2 2 -1 -1 0 0 0 0 orthogonal lifted from D6 ρ6 2 2 0 2 0 -1 2 2 2 2 -1 0 0 2 2 0 0 -1 -1 -1 -1 2 2 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ7 2 2 -2 0 0 2 2 0 -1-√5/2 -1+√5/2 2 -2 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D10 ρ8 2 2 2 0 0 2 2 0 -1+√5/2 -1-√5/2 2 2 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ9 2 2 -2 0 0 2 2 0 -1+√5/2 -1-√5/2 2 -2 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D10 ρ10 2 2 2 0 0 2 2 0 -1-√5/2 -1+√5/2 2 2 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ11 3 3 1 1 -1 0 -1 -3 3 3 0 -1 1 3 3 1 1 0 0 0 0 -1 -1 0 0 -1 -1 -1 -1 orthogonal lifted from C2×S4 ρ12 3 3 1 -1 1 0 -1 3 3 3 0 -1 -1 3 3 1 1 0 0 0 0 -1 -1 0 0 -1 -1 -1 -1 orthogonal lifted from S4 ρ13 3 3 -1 -1 -1 0 -1 3 3 3 0 1 1 3 3 -1 -1 0 0 0 0 -1 -1 0 0 1 1 1 1 orthogonal lifted from S4 ρ14 3 3 -1 1 1 0 -1 -3 3 3 0 1 -1 3 3 -1 -1 0 0 0 0 -1 -1 0 0 1 1 1 1 orthogonal lifted from C2×S4 ρ15 4 -4 0 0 0 -2 0 0 4 4 2 0 0 -4 -4 0 0 0 0 -2 -2 0 0 2 2 0 0 0 0 orthogonal lifted from C4.3S4 ρ16 4 4 0 0 0 -2 4 0 -1-√5 -1+√5 -2 0 0 -1+√5 -1-√5 0 0 0 0 1+√5/2 1-√5/2 -1-√5 -1+√5 1-√5/2 1+√5/2 0 0 0 0 orthogonal lifted from S3×D5 ρ17 4 4 0 0 0 -2 4 0 -1+√5 -1-√5 -2 0 0 -1-√5 -1+√5 0 0 0 0 1-√5/2 1+√5/2 -1+√5 -1-√5 1+√5/2 1-√5/2 0 0 0 0 orthogonal lifted from S3×D5 ρ18 4 -4 0 0 0 1 0 0 4 4 -1 0 0 -4 -4 0 0 √3 -√3 1 1 0 0 -1 -1 0 0 0 0 orthogonal lifted from C4.3S4 ρ19 4 -4 0 0 0 1 0 0 4 4 -1 0 0 -4 -4 0 0 -√3 √3 1 1 0 0 -1 -1 0 0 0 0 orthogonal lifted from C4.3S4 ρ20 4 -4 0 0 0 -2 0 0 -1-√5 -1+√5 2 0 0 1-√5 1+√5 0 0 0 0 1+√5/2 1-√5/2 0 0 -1+√5/2 -1-√5/2 ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 ζ87ζ54-ζ87ζ5+ζ85ζ54-ζ85ζ5 ζ83ζ54-ζ83ζ5+ζ8ζ54-ζ8ζ5 orthogonal faithful ρ21 4 -4 0 0 0 -2 0 0 -1-√5 -1+√5 2 0 0 1-√5 1+√5 0 0 0 0 1+√5/2 1-√5/2 0 0 -1+√5/2 -1-√5/2 -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 ζ83ζ54-ζ83ζ5+ζ8ζ54-ζ8ζ5 ζ87ζ54-ζ87ζ5+ζ85ζ54-ζ85ζ5 orthogonal faithful ρ22 4 -4 0 0 0 -2 0 0 -1+√5 -1-√5 2 0 0 1+√5 1-√5 0 0 0 0 1-√5/2 1+√5/2 0 0 -1-√5/2 -1+√5/2 ζ83ζ54-ζ83ζ5+ζ8ζ54-ζ8ζ5 ζ87ζ54-ζ87ζ5+ζ85ζ54-ζ85ζ5 ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 orthogonal faithful ρ23 4 -4 0 0 0 -2 0 0 -1+√5 -1-√5 2 0 0 1+√5 1-√5 0 0 0 0 1-√5/2 1+√5/2 0 0 -1-√5/2 -1+√5/2 ζ87ζ54-ζ87ζ5+ζ85ζ54-ζ85ζ5 ζ83ζ54-ζ83ζ5+ζ8ζ54-ζ8ζ5 -ζ83ζ53+ζ83ζ52-ζ8ζ53+ζ8ζ52 ζ83ζ53-ζ83ζ52+ζ8ζ53-ζ8ζ52 orthogonal faithful ρ24 6 6 2 0 0 0 -2 0 -3-3√5/2 -3+3√5/2 0 -2 0 -3+3√5/2 -3-3√5/2 -1+√5/2 -1-√5/2 0 0 0 0 1+√5/2 1-√5/2 0 0 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from D5×S4 ρ25 6 6 -2 0 0 0 -2 0 -3+3√5/2 -3-3√5/2 0 2 0 -3-3√5/2 -3+3√5/2 1+√5/2 1-√5/2 0 0 0 0 1-√5/2 1+√5/2 0 0 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5×S4 ρ26 6 6 2 0 0 0 -2 0 -3+3√5/2 -3-3√5/2 0 -2 0 -3-3√5/2 -3+3√5/2 -1-√5/2 -1+√5/2 0 0 0 0 1-√5/2 1+√5/2 0 0 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from D5×S4 ρ27 6 6 -2 0 0 0 -2 0 -3-3√5/2 -3+3√5/2 0 2 0 -3+3√5/2 -3-3√5/2 1-√5/2 1+√5/2 0 0 0 0 1+√5/2 1-√5/2 0 0 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5×S4 ρ28 8 -8 0 0 0 2 0 0 -2+2√5 -2-2√5 -2 0 0 2+2√5 2-2√5 0 0 0 0 -1+√5/2 -1-√5/2 0 0 1+√5/2 1-√5/2 0 0 0 0 orthogonal faithful, Schur index 2 ρ29 8 -8 0 0 0 2 0 0 -2-2√5 -2+2√5 -2 0 0 2-2√5 2+2√5 0 0 0 0 -1-√5/2 -1+√5/2 0 0 1-√5/2 1+√5/2 0 0 0 0 orthogonal faithful, Schur index 2

Smallest permutation representation of GL2(𝔽3)⋊D5
On 80 points
Generators in S80
(1 54 12 69)(2 55 13 70)(3 51 14 66)(4 52 15 67)(5 53 11 68)(6 19 60 64)(7 20 56 65)(8 16 57 61)(9 17 58 62)(10 18 59 63)(21 34 78 47)(22 35 79 48)(23 31 80 49)(24 32 76 50)(25 33 77 46)(26 45 74 40)(27 41 75 36)(28 42 71 37)(29 43 72 38)(30 44 73 39)
(1 42 12 37)(2 43 13 38)(3 44 14 39)(4 45 15 40)(5 41 11 36)(6 21 60 78)(7 22 56 79)(8 23 57 80)(9 24 58 76)(10 25 59 77)(16 49 61 31)(17 50 62 32)(18 46 63 33)(19 47 64 34)(20 48 65 35)(26 67 74 52)(27 68 75 53)(28 69 71 54)(29 70 72 55)(30 66 73 51)
(16 23 49)(17 24 50)(18 25 46)(19 21 47)(20 22 48)(26 52 45)(27 53 41)(28 54 42)(29 55 43)(30 51 44)(31 61 80)(32 62 76)(33 63 77)(34 64 78)(35 65 79)(36 75 68)(37 71 69)(38 72 70)(39 73 66)(40 74 67)
(6 60)(7 56)(8 57)(9 58)(10 59)(16 23)(17 24)(18 25)(19 21)(20 22)(26 74)(27 75)(28 71)(29 72)(30 73)(36 53)(37 54)(38 55)(39 51)(40 52)(41 68)(42 69)(43 70)(44 66)(45 67)(61 80)(62 76)(63 77)(64 78)(65 79)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)
(1 46)(2 50)(3 49)(4 48)(5 47)(6 75)(7 74)(8 73)(9 72)(10 71)(11 34)(12 33)(13 32)(14 31)(15 35)(16 44)(17 43)(18 42)(19 41)(20 45)(21 68)(22 67)(23 66)(24 70)(25 69)(26 56)(27 60)(28 59)(29 58)(30 57)(36 64)(37 63)(38 62)(39 61)(40 65)(51 80)(52 79)(53 78)(54 77)(55 76)

G:=sub<Sym(80)| (1,54,12,69)(2,55,13,70)(3,51,14,66)(4,52,15,67)(5,53,11,68)(6,19,60,64)(7,20,56,65)(8,16,57,61)(9,17,58,62)(10,18,59,63)(21,34,78,47)(22,35,79,48)(23,31,80,49)(24,32,76,50)(25,33,77,46)(26,45,74,40)(27,41,75,36)(28,42,71,37)(29,43,72,38)(30,44,73,39), (1,42,12,37)(2,43,13,38)(3,44,14,39)(4,45,15,40)(5,41,11,36)(6,21,60,78)(7,22,56,79)(8,23,57,80)(9,24,58,76)(10,25,59,77)(16,49,61,31)(17,50,62,32)(18,46,63,33)(19,47,64,34)(20,48,65,35)(26,67,74,52)(27,68,75,53)(28,69,71,54)(29,70,72,55)(30,66,73,51), (16,23,49)(17,24,50)(18,25,46)(19,21,47)(20,22,48)(26,52,45)(27,53,41)(28,54,42)(29,55,43)(30,51,44)(31,61,80)(32,62,76)(33,63,77)(34,64,78)(35,65,79)(36,75,68)(37,71,69)(38,72,70)(39,73,66)(40,74,67), (6,60)(7,56)(8,57)(9,58)(10,59)(16,23)(17,24)(18,25)(19,21)(20,22)(26,74)(27,75)(28,71)(29,72)(30,73)(36,53)(37,54)(38,55)(39,51)(40,52)(41,68)(42,69)(43,70)(44,66)(45,67)(61,80)(62,76)(63,77)(64,78)(65,79), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,46)(2,50)(3,49)(4,48)(5,47)(6,75)(7,74)(8,73)(9,72)(10,71)(11,34)(12,33)(13,32)(14,31)(15,35)(16,44)(17,43)(18,42)(19,41)(20,45)(21,68)(22,67)(23,66)(24,70)(25,69)(26,56)(27,60)(28,59)(29,58)(30,57)(36,64)(37,63)(38,62)(39,61)(40,65)(51,80)(52,79)(53,78)(54,77)(55,76)>;

G:=Group( (1,54,12,69)(2,55,13,70)(3,51,14,66)(4,52,15,67)(5,53,11,68)(6,19,60,64)(7,20,56,65)(8,16,57,61)(9,17,58,62)(10,18,59,63)(21,34,78,47)(22,35,79,48)(23,31,80,49)(24,32,76,50)(25,33,77,46)(26,45,74,40)(27,41,75,36)(28,42,71,37)(29,43,72,38)(30,44,73,39), (1,42,12,37)(2,43,13,38)(3,44,14,39)(4,45,15,40)(5,41,11,36)(6,21,60,78)(7,22,56,79)(8,23,57,80)(9,24,58,76)(10,25,59,77)(16,49,61,31)(17,50,62,32)(18,46,63,33)(19,47,64,34)(20,48,65,35)(26,67,74,52)(27,68,75,53)(28,69,71,54)(29,70,72,55)(30,66,73,51), (16,23,49)(17,24,50)(18,25,46)(19,21,47)(20,22,48)(26,52,45)(27,53,41)(28,54,42)(29,55,43)(30,51,44)(31,61,80)(32,62,76)(33,63,77)(34,64,78)(35,65,79)(36,75,68)(37,71,69)(38,72,70)(39,73,66)(40,74,67), (6,60)(7,56)(8,57)(9,58)(10,59)(16,23)(17,24)(18,25)(19,21)(20,22)(26,74)(27,75)(28,71)(29,72)(30,73)(36,53)(37,54)(38,55)(39,51)(40,52)(41,68)(42,69)(43,70)(44,66)(45,67)(61,80)(62,76)(63,77)(64,78)(65,79), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75)(76,77,78,79,80), (1,46)(2,50)(3,49)(4,48)(5,47)(6,75)(7,74)(8,73)(9,72)(10,71)(11,34)(12,33)(13,32)(14,31)(15,35)(16,44)(17,43)(18,42)(19,41)(20,45)(21,68)(22,67)(23,66)(24,70)(25,69)(26,56)(27,60)(28,59)(29,58)(30,57)(36,64)(37,63)(38,62)(39,61)(40,65)(51,80)(52,79)(53,78)(54,77)(55,76) );

G=PermutationGroup([[(1,54,12,69),(2,55,13,70),(3,51,14,66),(4,52,15,67),(5,53,11,68),(6,19,60,64),(7,20,56,65),(8,16,57,61),(9,17,58,62),(10,18,59,63),(21,34,78,47),(22,35,79,48),(23,31,80,49),(24,32,76,50),(25,33,77,46),(26,45,74,40),(27,41,75,36),(28,42,71,37),(29,43,72,38),(30,44,73,39)], [(1,42,12,37),(2,43,13,38),(3,44,14,39),(4,45,15,40),(5,41,11,36),(6,21,60,78),(7,22,56,79),(8,23,57,80),(9,24,58,76),(10,25,59,77),(16,49,61,31),(17,50,62,32),(18,46,63,33),(19,47,64,34),(20,48,65,35),(26,67,74,52),(27,68,75,53),(28,69,71,54),(29,70,72,55),(30,66,73,51)], [(16,23,49),(17,24,50),(18,25,46),(19,21,47),(20,22,48),(26,52,45),(27,53,41),(28,54,42),(29,55,43),(30,51,44),(31,61,80),(32,62,76),(33,63,77),(34,64,78),(35,65,79),(36,75,68),(37,71,69),(38,72,70),(39,73,66),(40,74,67)], [(6,60),(7,56),(8,57),(9,58),(10,59),(16,23),(17,24),(18,25),(19,21),(20,22),(26,74),(27,75),(28,71),(29,72),(30,73),(36,53),(37,54),(38,55),(39,51),(40,52),(41,68),(42,69),(43,70),(44,66),(45,67),(61,80),(62,76),(63,77),(64,78),(65,79)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75),(76,77,78,79,80)], [(1,46),(2,50),(3,49),(4,48),(5,47),(6,75),(7,74),(8,73),(9,72),(10,71),(11,34),(12,33),(13,32),(14,31),(15,35),(16,44),(17,43),(18,42),(19,41),(20,45),(21,68),(22,67),(23,66),(24,70),(25,69),(26,56),(27,60),(28,59),(29,58),(30,57),(36,64),(37,63),(38,62),(39,61),(40,65),(51,80),(52,79),(53,78),(54,77),(55,76)]])

Matrix representation of GL2(𝔽3)⋊D5 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 240 0 0 0 0 0 0 240 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 240 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 240 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 1 0 0
,
 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 240
,
 0 1 0 0 0 0 240 189 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 16 173 0 0 0 0 64 225 0 0 0 0 0 0 99 99 99 0 0 0 99 142 0 99 0 0 99 0 142 142 0 0 0 99 142 99

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,240,0,0,0,0,0,0,240,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,240],[0,240,0,0,0,0,1,189,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,64,0,0,0,0,173,225,0,0,0,0,0,0,99,99,99,0,0,0,99,142,0,99,0,0,99,0,142,142,0,0,0,99,142,99] >;

GL2(𝔽3)⋊D5 in GAP, Magma, Sage, TeX

{\rm GL}_2({\mathbb F}_3)\rtimes D_5
% in TeX

G:=Group("GL(2,3):D5");
// GroupNames label

G:=SmallGroup(480,970);
// by ID

G=gap.SmallGroup(480,970);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1680,3389,93,1347,2111,3168,172,1272,1909,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=c^3=d^2=e^5=f^2=1,b^2=a^2,b*a*b^-1=f*a*f=d*b*d=a^-1,c*a*c^-1=a*b,d*a*d=f*b*f=a^2*b,a*e=e*a,c*b*c^-1=a,b*e=e*b,d*c*d=c^-1,c*e=e*c,f*c*f=a*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

Export

׿
×
𝔽